Since \(g(u, u) = -1\)  (by definition of a perfect fluid), continuity implies \(g_\text{{\rm ext}}(\widetilde{u}, \widetilde{u}) = -1\)  at \(\mathcal {O}\) . Using [1]} and applying the Gram-Schmidt orthogonalization process appropriately, for any \(0 < \varepsilon <1\) , we can assume that the coordinates \((x^0, \cdots , x^n)\)  on \(U\)  satisfy
